Dual porosity DRM formulation for flow and transport through fractured porous media

Transcription

1 Boundary Eleents XXVII 407 Dual porosity DRM orulation or lo and transport through ractured porous edia T. aardzioska & V. Popov Wessex Institute o Technology, UK Abstract The ain objective o this ork is to develop a odel or lo and solute transport in ractured porous edia based on the dual porosity approach and the Dual Reciprocity Method. The developed odel is copared to the results obtained using the equivalent continuu odel and the discrete racture/nonhoogeneous odel. The coparison is perored or a square porous doain ith regular esh o three parallel ractures cobined ith a urther three ractures perpendicular to the irst ones. The results helped to dra soe conclusions in respect to the siilarity o potentials as ell as luxes or the dierent ethods. In this research the Boundary Eleent Dual Reciprocity Method schee (BE DRM) has been used, in cobination ith the dual-porosity odel. Keyords: ractured porous edia, dual porosity odel, DRM, BEM. Introduction Fractured porous edia could be odelled ith a non-hoogeneous odel here porosity and pereability are alloed to vary discontinuously and rapidly, as both quantities are signiicantly larger in the ractures than in the porous rock. This approach is considered to be ore accurate than the dual porosity odel, since it introduces a saller nuber o sipliications or approxiations. Hoever, the coputational and data requireents or treating such a odel are oten too large, hich akes this approach not suitable or every practical case. As an alternative, the discontinuous nature o the pereability and porosity can be avoided by replacing the by averaged values. uch sipliication is used in the equivalent continuu odel [], hich does not contain ractures,.itpress.co, IN X (on-line)

2 408 Boundary Eleents XXVII but the results ay be quite dierent ro the ones obtained using the nonhoogeneous odel or a given geoetry and properties o the racture netork. A progress in the continuu odel approach has been achieved ith dual porosity odels [2]. The odel assues that the ractured porous edia is a continuu consisting o to overlapping regions [3]. The coupling ter beteen the to pore systes as derived, aong the others, by Gerke and van Genuchten [4, 5], and this approach has been olloed in this ork. The nuerical approach used is based on the Boundary Eleent Dual Reciprocity Method Multi Doain schee (BE DRM-MD) [6, 7]. 2 The dual porosity lo and transport odel or ractured porous edia 2. Dual porosity odel Flo and transport in structured porous edia are requently described using dual-porosity (or double-porosity) odels. In such a odel the void space o the ractures is considered as a continuu hile the void space ithin the blocks is regarded as another continuu. The dual-porosity ediu is considered to be a superposition o these to systes over the sae volue. The to pore systes interact by exchanging ater and solutes in response to pressure head and concentration gradients. Macroscopically, to lo velocities, to pressure heads, and to solute concentrations characterize the porous ediu at any point in tie and space. Dual-porosity (DP) odels assue that both ater lo and solute transport can be described by to equations, hich are coupled through a ter that describes the exchange o luid or solute beteen the to pore regions. ince lo in the ractures is uch ore rapid, the early dual porosity odels neglected the lo through the atrix block syste [2]. The odel used in this ork is based on the one o Gerke and van Genuchten [4, 5] that solves the ull odels in both systes, porous atrix and racture netork. 2.2 Equations or lo Assuing applicability o Darcy s la, saturated ater lo in ractured porous edia is described by a coupled pair o equations [4, 5, 8, 9]: h 2 Γ C = K h (a) t h 2 Γ C = K h + (b) t here h is pressure head [L], C is speciic storativity [L - ], K is hydraulic conductivity [LT - ], t is tie [T] and is relative voluetric proportion o the racture pore syste. The paraeter Γ is the ater transer ter (T - ), and is given as.itpress.co, IN X (on-line)

3 ( h h ) Γ = α (2) here α is irst-order ass transer coeicient or lo [L - T - ] and is given as * β α = α K a = γ 2 K a (3) a here a is hal idth o the atrix block, or distance ro the centre o the ictitious atrix block to the racture boundary [L], β is diensionless actor depending on the geoetry o the aggregates and usually takes values beteen 3, or rectangular slabs, and 5 or spheres, γ is 0.4 and is relatively independent o the aggregate geoetry and the applied initial pressure and conditions, and K a is eective hydraulic conductivity o the atrix at the racture/atrix interace and is deined in the present case as Ka = 0.5 K a + Ka (4) ubstituting expression (2) or Γ into (7) and considering that = - yields: 2.3 Equations or transport ( h h ) C h 2 α h = + K t K ( h h ) 2 C h α h = + K t K Boundary Eleents XXVII 409 (5a) (5b) In the siilar anner as or the lo, the solute transport in a saturated ractured porous ediu is described using to coupled dual-porosity advectiondispersion equations: 2 c c c Γ c V V s = + + x + (6a) D t x y y 2 c c c c V V Γ s = + + x D t x y (6b) y here c is solute concentration [ML -3 ], D is dispersion coeicient [L 2 T - ], V x and V y are velocities in the ractures in x and y directions [MT - ], respectively, V x and V y are velocities in the atrix blocks in x and y directions [MT - ], respectively, and Γ s is solute ass transer ter [ML -3 T - ] given as Γc 0 ( Γ Γ s = αs )( c c) + (7) Γ c Γ 0.itpress.co, IN X (on-line)

4 40 Boundary Eleents XXVII β here α s = 2 D a is irst-order solute ass transer coeicient [T - ], Γ is ater a transer ter, and D a is eective diusion coeicient [L 2 T - ], calculated as average value or the to regions. 3 The nuerical ipleentation In the exaples shon in this ork, linear variation o the ield variables is assued over the boundary eleents. 3. Flo odel The DRM representations o (a) and (b) are shon belo h α( h h ) Hh Gq = C + (8a) t h α( ) h h Hh Gq = C + (8b) t here the olloing notation is used i = ( HU GQ) F (9) K i Next a inite dierence approxiation or the tie derivative and a linear variation o u and q ithin each tie-step are introduced yielding α C n+ n+ H θ h h Gθqq t = α C α n n = H ( θ h) h + G( θq) q h t α C n+ n+ H θ h h Gθqq = t α C n n ( H ) h G( α = θ h + θq) q h t (0) ().itpress.co, IN X (on-line)

5 Boundary Eleents XXVII Transport odel The DRM representations o (6a) and (6b) are shon belo here: H c G q c c c = + V + V x t x y y + αs ( c c) + Γ c c c c H c G q V V = + + x t x y y ( ) αs c c + Γ c i = ( HU GQ) F D i (2a) (2b) (3) Next, the partial derivatives are represented using the usual DRM approxiation [0], the tie derivative is represented using inite dierence approxiation and c and q are represented by using a linear variation ithin each tie-step, yielding ( ) θ + c θ + q ( )( θc) n n H P c G q H P c n t = t α s ( n + G θq) q c (4) + + ( ) θ θ ( )( θ ) n n n H P c c G qq = H P c c t t (5) n Γ G( θ + q) q αs + c here: F F α s P V F V F Γ = x (6) x y y F F P = V F + V F + α x y s x y (7).itpress.co, IN X (on-line)

6 42 Boundary Eleents XXVII The coplete lo and transport proble involves our variables or each o the to doains, ractures and porous block, consisting o one hydraulic head, one lo lux, one solute concentration and one solute lux. 3.3 olution procedure or the DP odel The syste o equations (0) and (), hich also applies to the syste equations (4) and (5), can be solved siultaneously, as shon in (8), or by using an iterative schee, as shon in Table. In (8) BC represents boundary conditions and u represents hydraulic head or solute concentration. In the present study the iterative schee as selected as a better choice as in this ay the possibility o obtaining an ill conditioned syste o equations due to large dierence in the hydraulic conductivities, or advective ters, in the to dierent porous edia, ractures and porous atrix, is avoided. n+ n u u 0 q F 0 q = + 0 [ MB] u 0 MB u q q [ F] [ BC] (8) 4 Nuerical exaples In this exaple a square area is analysed ith diensions 0.46 x , here 24.4% o the volue is occupied by ractures []. To dierent eshes ere analysed, one ith horizontal and vertical ractures, in this exaple reerred to as the original esh, and the other one ith rotated esh o ractures under 45 o angle in respect to the original esh. In the both cases care has been taken or the analysed parts o the doains corresponding to porous edia and ractures to be equivalent in ters o volue, and also the utual distance beteen the ractures to be the sae. The nuerical schee has been tested and has shon good agreeent toards a D analytical solution or hoogeneous doain, prior to solving this exaple. The dual porosity odel paraeters here deined as: hal-distance beteen the ractures a=0.06, voluetric actor =24.4%, geoetry o the atrix blocks β=3.0, γ =0.4, and eective hydraulic conductivity K a taken as an average value o both conductivities in the atrix block K = /d, and in the ractures K = /d. The value o the speciic storativity used or all the odels is Dispersion coeicients D = /d and D = /d ere used or the transport siulation. The boundary conditions or lo are: hydraulic head h (0,y,t) = h (0,y,t) =.05 at the inlet surace and h (L,y,t) = h (L,y,t) =.0 at the outlet surace, and noral derivatives q (x,0,t) = q (x,0,t) = 0 and q (x,b,t) = q (x,b,t) = 0 at the loer (y = 0) and the upper (y = b) boundaries o the doain, respectively. For the solute transport: concentration c (0,y,t) = c (0,y,t) =.2 and c (L,y,t) =.itpress.co, IN X (on-line)

7 Boundary Eleents XXVII 43 c (L,y,t) =.0, and zero noral derivative q at y = 0 and y = b are iposed. Here, L is the length o the doain in the direction o the lo and b is its idth. The initial conditions in the ractures and the atrix pore syste are h(x,y,0) =.0 or the lo and c(x,y,0) =.0 or the transport. Table : The iterative schee used (BC boundary conditions; IC initial conditions). Iteration u, q racture netork (irst step in each iteration) (), ( ) ( BC+ IC) (), ()( BC+ IC) ( + ) ( + ) u, q atrix block (second step in each iteration) ( ) u t u t u t + t, u( t) q t q t q ( t + t), q( t) u t t u q t t q k k k+ k u ( ) ( ) k+ t + t, u t + t u t + t, u t + t k k k+ k q, q( t + t) q t + t, q t + t k + k + u u k + k + q q n+ n IF: u = u ( t + t) u ( t + t) 0 n+ n and u = u ( t + t) u( t + t) 0 THEN: stop the iterations ( ) ( ) ( ) ( ) In Figure 2 a coparison is shon beteen the results obtained using the present dual porosity (DP) odel and the previous results [] obtained using the equivalent continuu (EC) and non-hoogeneous (NH) odels or hydraulic head proiles in the ractures. The results obtained ith the three odels are in good agreeent. The coparison o estiated total lo luxes on the outlet o the doain is shon in Figure 3. The lux or the DP odel as calculated by using the voluetric actor o the ractures, hich is representative or the hole doain, in the igure reerred to as DP (), or by using the exact aperture o the ractures participating in the considered cross section, in the igure reerred to as DP. It is apparent that both NH solutions obtained using to dierent eshes.itpress.co, IN X (on-line)

8 44 Boundary Eleents XXVII and the DP odel using the exact aperture o the ractures participating in the considered cross section sho lux results that are in good agreeent. atrix block racture zone a) original esh b) rotated esh atrix block racture zone Figure : quare esh ith discrete racture zones: a) original esh; b) rotated esh hydraulic head NH orig; r NH rot; r EC DP; r x [] Figure 2: Hydraulic head proiles inside the ractures estiated using the three odels (t=0.00days; α =.454, =0.244, K =0-7 /s, K =0-9 /s, C =C =0-4 - ) Figure 4 shos the concentration proiles in the ractures and atrix blocks obtained ith the three dierent odels. The coparison o estiated total solute luxes on the outlet o the doain shoed siilar behaviour as the ones or the lo..itpress.co, IN X (on-line)

9 Boundary Eleents XXVII lux NH orig NH rot DP () DP EC t [days] Figure 3: Total lo luxes at outlet surace or the three odels; α=.454, =0.244, K =0-7 /s, K =0-9 /s, C =C= concentration NH orig; r NH orig; b EC NH rot; r DP r DP b x [] Figure 4: Concentration proiles or the three odels (t=0.0 days; αs=22.9, =0.244, D =0.05 2/day, D = /day) 5 Conclusions A odel as developed or lo and solute transport in ractured porous edia based on the dual porosity approach and the Dual Reciprocity Method. The ill conditioning o the syste atrix due to very large dierence in the pereabilities o the ractures and porous atrix is avoided by solving the equations in the porous atrix and ractures through an iterative schee. The developed odel as validated using analytical solutions or siple D cases and then copared to the results previously obtained using the equivalent continuu odel and the discrete racture/non-hoogeneous odel [], shoing good agreeent or both, hydraulic head and concentration proiles in the doain. The results sho that the luxes should be evaluated ith special care hen the DP odel is used. While the NH odel can accurately estiate the luxes.itpress.co, IN X (on-line)

10 46 Boundary Eleents XXVII since the total cross section o the ractures hich participate in the considered cross section o the porous edia is accurately estiated, in the case o the DP odel in a general case the luxes ould be estiated by using the voluetric actor o the ractures, hich introduces errors. The proble is that in such case, not only the active ractures, hich contribute toards the lux, are taken into account, but also the ractures ith stagnant ater, providing a signiicant overestiate o the lux. The EC odel shos the sae probles hen calculating luxes, but or the case o the EC odel used in [] it is ore diicult to iprove the results, as the error is inherent in the solution o the EC odel itsel and is due to the ay that the equivalent edia properties are estiated. Reerences [] Berkoitz, B., Bear J., Braester, C. Continuu odels or containant transport in ractured porous orations. Water Resources Research, 24 (988), [2] Barenblatt GI, Zheltov IP, Kochina IN. Basic concepts in the theory o hoogeneous liquids in issured rocks. J Appl Math Mech (Engl Transl) 960; 24: [3] Warren JE, Root PJ. The behaviour o naturally ractured reservoirs. ociety o Petroleu Engineers Journal 963;3: [4] Gerke HH, van Genuchten MT. A dual-porosity odel or siulating the preerential oveent o ater and solutes in structured porous edia. Water Resources Research 993; 29(2): [5] Gerke HH, van Genuchten MT. Evaluation o a irst-order ater transer ter or variably saturated dual-porosity lo odels. Water Resources Research 993; 29(4): [6] Popov V., and H. Poer, DRM-MD approach or the nuerical solution o gas lo in porous edia, ith application to landill, Eng. Anal. Boun. Ele., 23, 75-88, 999. [7] Popov V., and H. Poer, The DRM-MD integral equation ethod: An eicient approach or the nuerical solution o doain doinant probles, Int. J. Nu. Meth. Engrg, 44, , 999. [8] Dykhuizen RC. A ne coupling ter or dual-porosity odels. Water Resources Research 990; 26(2): [9] Gerke HH, van Genuchten MT. Macroscopic representation o structural geoetry or siulating ater and solute oveent in dual-porosity edia. Advances in Water Resources 996; 9(6): [0] Partridge PW, Brebbia CA, Wrobel LC. The dual reciprocity boundary eleent ethod. outhapton UK: Coputational Mechanics Publications; 992. [] aardzioska, T., Popov, V. (2003), DRM-MD or lo and transport through ractured rocks, in: Boundary Eleents XXV, Edt. Brebbia, C.A., Poljak, D., Roje, V., WIT Press, outhapton, itpress.co, IN X (on-line)